Exercise 18.4 (f) of Kosniowski's "A first course in Algebraic Topology" reads
Let $p: (\tilde{X}, \tilde{x}_0) \rightarrow (X, x_0)$ be a covering. Suppose the fundamental group of $X$ at $x_0$ is $\pi_1(X, x_0) = \mathbb{Z}$ and $p^{-1}(x_0)$ is finite. Find the fundamental group of $\tilde{X}$ at $\tilde{x}_0$.
Now all I can say is that $\tilde{X}$ cannot be simply connected, else there would be a 1-to-1 correspondence between $\mathbb{Z}$ and $p^{-1}(x_0)$, which is not possible with the latter finite.
I think I should somehow use the homomorphism $p_{\ast}$ induced by the covering map, but I have no further idea how to procede from here. Any suggestion?
Best Answer
Hint: $p_*$ must be an injective group homomorphism from $\pi_1(\tilde X)$ to $\Bbb Z$.